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After hours of celebration on Wembley Way, footage emerged of fans breaking down security gates and rushing past guards to gain access to the stadium.
Fighting was then reported on the concourses while fans who had paid for tickets found their seats occupied as supporters filled stairways and exits.
The Met Police later confirmed after the match that 45 arrests had been made.
In a fierce message to those supporters, the FA deemed the behaviour as "entirely unacceptable" and will work to identify those guilty.
A statement read: "We strongly condemn the behaviour of a group of people that forced their way into Wembley Stadium before the EURO 2020 Final. This is entirely unacceptable.
"These people are an embarrassment to the England team and to all of the true fans who wanted to enjoy one of the most important matches in our history.
"We will work with the relevant authorities to take action against anyone who is identified to have illegally forced their way into the stadium."